Computers & Industrial Engineering

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1 Computers & Industrial Engineering 58 (2010) Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: A genetic algorithm approach for solving the daily photograph selection problem of the SPOT5 satellite Mohamed A.A. Mansour a, Maged M. Dessouky b, * a Industrial Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt b Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA , United States article info abstract Article history: Received 5 April 2008 Received in revised form 19 November 2009 Accepted 20 November 2009 Available online 26 November 2009 Keywords: Earth observation satellite Daily photograph selection Multi-criteria constrained optimization Genetic algorithms Genome coding Analysis of variance This article addresses the combinatorial optimization problem of managing earth observation satellites (EOSs) such as the French SPOT5, which is concerned with selecting on each day a subset of a set of candidate photographs. The problem has a significant economic importance due to its high initial investment cost that exists in these instruments and its solution difficulty resulting from the large solution space, making it an attractive research area. This article proposes a genetic algorithm (GA) for solving the SPOT5 selection problem using a new genome representation for maximizing not only a single objective as profit but a multi-criteria objective that includes the number of acquired photographs. Test results of our proposed GA show that it finds optimal solutions effectively for moderate size problems and obtains better results for two large benchmark instances coded 1403 and 1504 in the literature. Also, we verify the result that the best known value in the literature for problem coded 1401 is an optimal value. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The Earth Observing Satellites (EOSs) such as SPOT5 orbit the Earth to capture photographs and can collect data of almost any point of the planet. They have commercial and military applications (e.g., mapping, natural disaster management, etc.) since they can take detailed high-quality photographs of large areas. Effective mission planning is necessary in order to obtain a suitable return in the large capital investment of such systems. The mission planning problem of an EOS is concerned with maximizing a profit function subject to a set of satellite capacity constraints over the planning horizon. It can be defined as the selection of a subset of photographs from a candidate set in order to satisfy a maximal profit of the requested photographs. The problem contains a large number of decision variables and constraints making it a combinatorial optimization problem that is typically solved using heuristics. There are two types of photographs, mono and stereo, which a SPOT5 satellite can take and these photographs are taken by three different instruments (front, middle, and rear) located on the satellite. Mono photographs require only one camera which any of the three instruments can take and stereo photographs require two cameras, front and rear. Fig. 1a shows the process of acquiring photographs where a ground station can directly receive a photograph * Corresponding author. Tel.: ; fax: addresses: neuoneall@yahoo.com (M.A.A. Mansour), maged@usc.edu (M.M. Dessouky). if visible to the satellite or otherwise it has to be stored using onboard recorders until the satellite becomes visible again. Associated with each customer request is a profit or gain representing the customer value and a set of characteristics that defines the number of cameras needed to capture the photograph and the photograph s memory size. In practice, not all of the requested photographs can be taken due to operational constraints such as limited storage. The entry of SPOT5 into visible areas is dictated by its orbiting pattern so the capturing start time is fixed; consequently, the photograph capturing order cannot change, so it is actually a selection as opposed to a true scheduling problem according to Lemaître, Verfaillie, Jouhaud, Lachiver, and Bataille (2002) as shown in Fig. 1b. The Daily Photograph Selection Problem (DPSP) is a relatively new research problem with earlier work carried out by Verfaillie, Lemaître, and Schiex (1996) and Bensana, Verfaillie, Agnèse, Bataille, and Blumstein (1996). Verfaillie et al. (1996) introduced the Russian Doll Search (RDS) algorithm and used it to find the optimal solution for eight single orbit moderately-sized SPOT5 problems. They concluded that the RDS algorithm can usually outperform the common Depth First Branch and Bound (DFBB) algorithm, even if the latter uses the best dynamic variable orderings. A variable Valued Constraint Satisfaction Problem (VCSP) and an integer linear programming model of the problem was proposed by Bensana et al. (1996). They compared the performance of approximate solution methods (greedy search GS, tabu search TS) and exact solution methods (DFBB, RDS) with the solution /$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi: /j.cie

2 510 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) (a) SPOT5 orbit. (b) A photograph acquisition. Fig. 1. Photograph acquisition performed by SPOT5. found by a commercial optimization solver, CPLEX, using the DFBB rule. Due to the combinatorial nature of the problem, exact solution methods failed to find optimal solutions for large size problems and heuristics were necessary to solve the larger problems in order to find good approximate solutions. Bensana, Lemaître, and Verfaillie (1999) presented 20 benchmark problems which can be downloaded from ftp://ftp.cert.fr/ pub/lemaitre/lvcsp/pbs/spot5.tgz. These 20 benchmark instances were randomly generated from a SPOT5 order book and consist of two types of problems, single and multi-orbit instances, as depicted in Table 1. The first three columns of the table list the problem instance, number of photographs (n), and number of capturing conflicts (nc) of the problem respectively. Problem codes less than a 1000 indicate a single orbit problem without capacity constraints while the others have multiple orbits with capacity constraints. The subsequent columns provide the solution to the maximum profit problem for different solution approaches used in the literature. When the profit is italicized in the table, it means that the optimal profit was found for that problem instance. Optimal solutions for 14 of the test problems were found by either using CPLEX, a Valued Constraint Satisfaction Problems (VCSP) formulation (Schiex, Fargier, & Verfaillie, 1995), or a RDS algorithm (Verfaillie et al., 1996). Approximate solutions for the other six problems were found by a TS procedure (Bensana et al., 1996; Vasquez & Hao, 2001). Gabrel and Murat (2003) derived two upper bound procedures, one based on graph theory and the other on a column generation technique on a vertex-path formulation, and tested them on 10 single orbit selection problems. Vasquez and Hao (2001) formulated the problem as a generalized version of the well-known knapsack model. They developed a TS algorithm which integrates some important features including an efficient neighborhood, a dynamic tabu tenure mechanism, and techniques for constraint handling, intensification and diversification. Their TS procedure found optimal solutions for all single orbit problems and the equivalent best known solution as in Bensana et al. (1999) for several multiple orbit problems. Identifying good upper bounds was the focus of the work by Vasquez and Hao (2003). They used a partition-based approach (UPPB) where the problem is divided into sub-problems with each sub-problem being optimally solved by an iterative enumeration algorithm. They also used a column generation based approach to find an alternative upper bound (UPCG). The last upper bound that they derived is based on a linear program relaxation of the problem (UPLP). From the previous survey, we note that no attempt has been done to solve the SPOT5 selection problem using a genetic algorithm (GA) although a binary representation genome has been proposed for the TS procedure. Also, the prior research has focused on the single objective of maximizing the profit. In this paper we consider this single objective as well as the multi-criteria objective of maximizing profit and the number of captured photographs. GAs include features for handling a large number of constraints, modeling flexibility in dealing with problem complexity, utilizing less CPU memory allocation than mathematical programming techniques, handling the multi-criteria nature of real-world problems without any modeling complications, and ease of implementation. Furthermore, there have been a significant number of papers that have shown GAs effectiveness in solving hard combinatorial problems. For example, Beasley and Chu (1996) and Lorena and Lopes (1997) apply GA to the set covering problem. In summary, the two primary contributions of this paper to the SPOT5 literature are that: (1) we demonstrate the effectiveness of GA for solving the SPOT5 problem and show that it improves on the best known results for some of the test problems and (2) we expand the analysis to a multi-criteria formulation. The remainder of the paper is organized as follows: In the next section, the SPOT5 problem is described. Section 3 presents the details of the developed GA. Tests to fine-tune the parameters of the GA are presented in Section 4. Numerical results on the benchmark problems are presented in Section 5. Section 6 gives the conclusions and directions for future work. 2. Problem description We next present the photograph selection problem of the SPOT5 satellite based on the informal description by Bensana et al. (1999) and the reader is referred to Vasquez and Hao (2001) for an integer programming model formulation of the problem. They formulate the problem as a knapsack problem, which is known to be a NP-hard problem. There are two types of photographs, mono and stereo. The mono photographs require only one camera and any of the cameras (front, middle, and rear) can take the picture. The stereo photographs require two cameras, front and rear. Associated with each photograph is its contribution to the profit if selected, the type of photograph (mono or stereo), and the amount of storage memory

3 Table 1 Summary results of the 20 SPOT5 Benchmark problems. Problem characteristics Verfaillie et al. (1996) Bensana et al. (1996) Bensana et al. (1999) Gabrel and Murat (2003) Vasquez and Hao (2001) Vasquez and Hao (2003) Code n nc RDS BFBB DFBB RDS GR TS CPLEX, VCSP, RDS, TS UP TUP TS UPLP UPCG UPPB ,032 12,032 12,032 12,032 12,032 12, , , , , , , , , , ,053 53,053 56,053 50,053 56,053 56, , , ,102 16,102 15,078 16,102 16,097 16,101 16, , , ,120 22,118 21,096 22,120 22,112 22,116 22,120 22, , ,100 13,100 12,088 13,100 12,102 13,100 13,100 22, , , ,137 15,137 13,101 15,137 15,129 15,136 15, , , ,125 19,123 19,104 19,125 19,116 19,123 19,125 19, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,158 60,155 61,158 61,158 61,158 61,158 61, , , , , , , , , , , , , , ,294 BFBB: The best first branch and bound embedded in CPLEX 4.0 (time limit 1800 s). DFBB: The standard depth first branch and bound (time limit of 600 s per sub-problem). RDS: The Russian Dolls Search (time limit 1800 s). GR: The multi-greedy algorithm. TS: The tabu search. UP: An upper bound using the graph theory. TUP: A tight upper bound with the vertex-path formulation and column generation procedure. UPLP: An upper bound based on LP relaxation. UPCG: An upper bound based column generation. UPPB: A partition-based upper bound approach. Italic indicates that the optimal profit was found. M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010)

4 512 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) that the photograph requires. Out of the possible candidate set S of mono or stereo photographs that can be taken, the problem is to select a subset S 0 that maximizes the profit. We also consider a multiple objective problem of maximizing profit and the total number of acquired photographs. The total number of acquired photographs is a surrogate measure for customer satisfaction since more demand that is met leads to higher customer satisfaction. There are a number of constraints associated with the problem. First, there are three cameras (front, rear, and middle) on board the satellite. Each camera can only acquire a single picture at each instant of time and there is a minimal transition time between pictures. Based on the memory requirements of each photograph, there is an on-board recording capacity constraint limiting the amount of stored photographs. 3. A genetic algorithm for the DPSP Genetic algorithms (GAs) are optimization heuristics based on stochastic search, which mimics the biological process of natural selection (Davis, 1991; Goldberg, 1989). They have been used to solve a wide variety of combinatorial optimization problems such as traveling salesman, vehicle routing, scheduling, and layout design. GAs have been used to effectively solve a large number of complex problems that deal with multiple objectives and constraints. The GA begins with an initial population. The population is made of a set of genomes (set of feasible solutions) and it continues to evolve until a specified stopping criterion is reached. At each generation, new genomes are mated as a result of applying genetic operators such as crossover and mutation and the good genomes replace the worst ones in the current generation (Michalewicz, 1992). In spite of GAs wide use, no research work to date has been done concerning its application to the DPSP of SPOT5 satellites. The most widely used local search technique to this problem has been the TS algorithm (Bensana et al., 1999). The GA variation that we use in this paper is a steady-state GA which uses overlapping populations where the newly generated individuals replace a portion of the population in each generation. We next describe the six operators controlling the GA s behavior. These operators are representation, initialization, selection, crossover scheme (type and probability), mutation scheme (type and probability), and replacement percentage operators. The population size and the number of generations vary according to the target problem complexity so they are experimentally determined in this article Representation The proposed genome representation consists of a number of genes equal to the total number of candidate photographs; each gene holds alleles of 0, 1, 2, 3 for mono photographs and 0, 13 for stereo ones. An allele value of 0 represents a photograph is not selected to be taken and any other value (1, 2, 3, 13) represents the satellite s camera assigned to capture the photograph. For example, consider a SPOT5 selection problem coded 8 in the test set that consists of eight photographs; each one can be captured by a specific camera type and a photograph s option is constrained by a set of forbidden camera/photograph pairs of values. The first four photographs are mono and the last four are stereo. The genome representation depicted in Fig. 2 represents 10 genomes each with 8 genes. The first one corresponds to assigning photographs 1, 2, 3, and 6 to be captured since they have nonzero allele values. Photographs 1 and 2 are captured by camera 1, photograph 3 by camera 2, and photograph 6 by camera 13, respectively. Not all the generated genomes in the initial population or after performing the genetic operators are feasible. Column P provides the profit gain for feasible genomes and is equal to zero for infeasible ones. We will handle the genome s infeasibility in the objective evaluation part comprehensively and illustrate how columns gm, f, and Gscore can be calculated. The proposed genome has advantages over a binary representation. The advantages are: a. With a binary representation, a problem of eight photographs requires 3 8 = 24 genes each with a 0 1 allele compared with eight genes for the proposed genome. b. Each gene holds only a possible photograph/camera solution so there is no schedule conflict in the gene, whereas in binary genomes each three genes representing a photograph/ camera solution has a high possibility of not being feasible with respect to the photograph. c. With a binary representation, the search sample space is 2 24 compared to 5 8 in the proposed genome Initialization We first sort set S into two subsets A and B. Set A contains photographs that do not have logical restrictions on other photographs. Set B is the difference between S and A. Each set has an initialization scheme as shown in Fig Initialization for set A photographs Set A contains the photographs that do not logically forbid other photographs from being selected. Their only restriction on other photographs is consumption of memory and thereby reducing the available memory for other photographs. Divide A into two subsets C and D where C includes zero memory photographs regardless of their profit value and D = A C. Fix all C element s locus by the photograph option to indicate its selection. For each element in D, calculate the profit/memory ratio and then sort D with respect to this ratio in descending order. Take the appearance probability in the genome of the first 5 photographs in D to be equal to 0.99, the next 5 equal to 0.98, and the remainder equal to 0.95 at the corresponding locus. Photograph no gm f Gscore P Genome number Fig. 2. The genome representation for DPSP.

5 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) Start Initialization operator sort set S into two subsets A and B A contains photographs that do not have logical restrictions on other photographs and B is the difference between S and A For each photograph in B, calculate the amount of profit forbidden if it is selected in the genome for each photograph option. Divide A to subsets C & D C: includes 0 memory size photographs D: includes A-C Fix all C element's locus by the photograph option to indicate its selection. For each element in D, calculate the profit/ memory ratio and then sort D with respect to this ratio in descending order. For each allele that has a selection probability 0.5, generate a random digit representing a photograph option with probability equal to [(forbidden profit for all photograph options -option s forbidden profit)/ 2*forbidden profit for all photograph options]. Select individuals for crossover Take the appearance probability in the genome of the first 5 photographs in D to be equal to 0.99, the next 5 equal to 0.98, and the remainder equal to 0.95 at the corresponding locus. These ranges increase the selection of high profit/ memory ratio photographs and decrease the effect of the knapsack constraint. Crossover individuals to produce offsprings Mutate the produced offsprings No Add cloned offsprings to current population forming a temporary one Remove worst individuals (replacement percentage) End Yes Stop? Fig. 3. Steps of the proposed GA. These ranges increase the selection of high profit/memory ratio photographs and decrease the effect of the knapsack constraint Initialization for set B photographs Set B includes photographs that forbid the capture of other photographs corresponding to the problem logical constraints. The initialization operator for B performs according to profit and memory size for each photograph option as follows: For each photograph in B, calculate the amount of profit forbidden if it is selected in the genome for each photograph option. The forbidden profit is the amount of profit that is lost in case Identity Arity Weight Linked Domain variables Option Possible values and memory requirements Memory Option Memory Option Memory (a) The ten photograph attributes. The forbidden pairs Fig. 4. A hypothetical SPOT5 problem. I II III Total memory available = 10 (b) The seven capturing constraints.

6 514 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) a photograph in set B is selected since the selection of this photograph makes the selection of some of the other photographs infeasible. For each allele that has a selection probability 0.5, generate a random digit representing a photograph option with probability equal to (1 (option s forbidden profit/forbidden profit for all photograph options)) Initialization example Fig. 4 represents a hypothetical SPOT5 problem with 10 photographs and 7 constraints with memory limitation. Fig. 4a represents at rows 1 3 the identity, weight, and domain size of each photograph. Photographs 0 3 have a domain of 3 since they are mono and any of the three cameras can capture them while photographs 4 9 have only one option since they must be captured by cameras 1 and 3. Rows 4 9 depict the possible capturing options and the required memory for the ten photographs. For example, column 1 represents a photograph coded by 0; its weight/profit equals to 10 and has three capturing options. It can be captured by cameras 1, 2, or 3 with a memory consumption of 2, 1, or 1, respectively. Fig. 4b describes the 7 capturing constraints. The first row gives the constraint s arity, the second and third link the conflicting photographs, and rows 4 9 give the forbidden pairs of capturing options. For example, the first column reads as arity equals to 2, it links photographs 1 and 0, and the forbidden capturing pairs of the photographs are (3, 3), (2, 2), and (1, 1). The total available memory equals to 10 as indicated at the bottom of Fig. 4b. The set S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and set A = {7, 8, 9} includes the set of photographs which do not conflict with other photographs as they do not exist in any cell of rows 2 or 3 of Fig. 4b. Subset A includes the 0-memory photographs (C) and the difference subset D = A C includes photographs that need memory allocation. In this example, subset C = {7} and subset D = {8, 9} since photograph 8 requires 5 units of memory and photograph 9 requires 10 units of memory. Each photograph element in set C has a probability of appearing in the genome of 1.0. Each photograph included in set D has a probability 0.99, 0.98, or 0.95 of appearing in the genome. The reason for selecting these probabilities is to let a chance for high-profit conflicting photographs in set B to appear in feasible genomes. If we do not choose to acquire photographs 8 and 9, a total of 15 memory units will be available to allocate to more profitable conflicting photographs that consume less than this memory amount. For each photograph in set D, calculate the ratio profit/memory and arrange this set in a descending order. The ratio for photograph 8 is 50/5 and for 9 is 50/ 10 so we arrange this set in the order 8 then 9. Since there are only two photographs in set D, they both have an appearance probability of If set D s photographs have more than one option, always take the option with the highest profit/memory ratio. Set B = S A = {0, 1, 2, 3, 4, 5, 6}, each element represents a photograph that may have more than one capturing option and different memory consumption. Each photograph in set B has an appearance probability of 0.5. Take photograph number 0, it has three options 1, 2, and 3 and has forbidden capturing pairs of (1, 1), (2, 2), and (3, 3) respectively. The corresponding forbidden profit is 1, 1, and 10. Therefore, take the probability of selecting options 1, 2, and 3 of photograph 0 as (12 1)/(2 12), (12 1)/ (2 12), and (12 10)/(2 12), respectively Genome score Our genome representation does not guarantee feasible genomes are maintained at each step of the GA. To discourage the selection of the infeasible genomes we add a cost term to the objective function to penalize infeasible genomes. The genome s score consists of two parts, genome satisfaction and objective value. The satisfaction part penalizes the genome s infeasibility due to the photograph conflict and memory constraints. The notations used in this article are listed below. am M ml n nic ni nic P gm 1 gm 2 gm f f 1 f 2 w 1 w 2 Gscore genome accumulated memory maximum profit gain from acquiring all candidate photographs EOS memory limit total number of photographs number of constraints number of taken photographs in the genome number of photograph conflicts genome profit genome photograph conflict measure genome memory constraint measure genome feasibility measure genome objective score for maximizing number of taken photographs and maximizing the profit genome objective score for maximizing number of taken photographs genome objective score for maximizing EOS profit weight of maximizing number of taken photographs weight of maximizing schedule profit genome overall score The conflict measure (gm) is the average of photograph conflict (gm 1 ) and memory satisfaction (gm 2 ) measures. gm 1 measures the genome conflict percentage and is equal to the difference between the problem s total number of photograph conflict constraints and Table 2 Experimental factors and levels. Factor Different factor levels Selection operator Tournament (TO), Ranking (RA), Roulette Wheel (RW), Uniform (UN) Crossover operator One Point (OPX), Two Points (TPX), Uniform (UX) Mutation operator Flip (FM), Gaussian (GM) Crossover probability 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 Mutation probability 0.001, 0.01, 0.019,..., 0.1 Replacement percentage 0.2, 0.25, 0.3,..., 0.5 Table 3 Analysis of variance. Source Sum. sq. d.f. Mean sq. F Prob. > F F F F F F F F1 F F1 F F1 F F1 F F1 F F2 F F2 F F2 F F2 F F3 F F3 F F3 F F4 F F4 F F5 F Error , Total ,095

7 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) Average fitness RW RA TO UN Selection operator Average fitness OPX TPX UX Crossover operator Average fitness FM Mutation operator GM Average fitness Crossover probability Average fitness Mutation probability Average fitness Replacement percentage Fig. 5. The sensitivity of GA s parameter settings for problem the actual number of photograph conflicts divided by the total number of photograph conflict constraints. The total number of conflict constraints is not equal to the total number of constraints since the latter includes capacity constraints. gm 1 equals 1 for feasible genomes and less than 1 for infeasible ones as shown in Eq. (1). gm 2 is measured as the ratio of the satellite memory limit over the genome accumulated memory of assigned photographs. gm 2 equals to 1 if the genome s accumulated memory is less than or equal to the memory limit or a ratio less than 1 as in Eq. (2). Feasible genomes must have gm 1 and gm 2 equal to 1, so the overall constraint satisfaction measure (gm) for genomes must equal to 1 if feasible. nc nic gm 1 ¼ nc 1 if am 6 ml gm 2 ¼ ml=am if am > ml gm ¼ gm 1 þ gm 2 ð3þ 2 The portion of the genome objective function for maximizing the number of taken photographs (f 1 ) is the number of acquired photographs divided by the total number of candidate photographs as in Eq. (4). For profit maximization, the genome objective (f 2 ) denotes the feasible genome profit divided by the maximum profit ð1þ ð2þ gain from acquiring all problem candidate photographs as shown in Eq. (5). For the multi-criteria of maximizing profit and number of taken photographs, f is the weighted average of f 1 and f 2 as in Eq. (6). The genome overall score (Gscore) equals the average of gm and f measures. f 1 ¼ ni n f 2 ¼ P M f ¼ w 1 f 1 þ w 2 f 2 w 1 þ w 2 Table 4 The number of generations and population size effect on GA s performance. Number of generations Population size , , , , , , ð4þ ð5þ ð6þ

8 516 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) Mean fitness Number of generations Fig. 6. The effect of population size and number of generations on GA s performance. Table 5 Results on SPOT5 benchmarks for maximizing profit and number of captured photographs. Problem code Best known, P GA s solution for maximizing, P GA s solution for maximizing, ni Pop_size/No_gen Best Mean Worst Rel Best Mean Worst Rel a 70 a a / a a 11,929 11, a / a a 91,661 83, a /350, a a 45,548 39, a /35, a 115 a a /65, a 49 a a /25, a 3082 a a /45, a a 13, a /40, a a 17,969 14, a /35, a 9096 a a /40, a a 11,832 10, a /40, a a 14,616 13, a /300, a a 18,308 17, a /300, , , ,971 15, /300, , , , , /50,00, , , , , /50,00, , , , , /60,00, a a 61,150 61, a /50, , , , , a /45,00, , , , , /50,00,000 Total 13,32,414 13,34,409 12,40, , Italic indicates that the solution is better than the best known solution in the published literature. a Indicates that the solution is optimal based on literature or CPLEX. 4. Genetic algorithm parameter settings There has been a significant amount of work on fine tuning the parameters of the GA. However, most of these findings are problem specific. There are a number of different ways to select a solution and perform crossover and mutation operators. We evaluate the suitability of the different ways to perform these operators for the DPSP next. In order to identify the best GA parameter settings for the DPSP, we conducted a full experimental design on six factors to maximize f. The six factors were tested on a moderate size test problem No. 503 (143 photographs, 492 constraints). The objective of these experiments is to define the most effective factor combinations and to determine which factors or combinations of factors are associated with the difference in the GA s behavior. The factors are selection, crossover, mutation schemes, crossover and mutation probabilities, and replacement percentage symbolized as F1 F6 respectively. Table 2 shows the levels that were studied for each factor. For the selection operator (F1), four levels were tested (Tournament, Ranking, Roulette Wheel, and Uniform). For the crossover operator (F2), three levels were tested (One Point, Two Points, and Uniform) and for the mutation operator (F3) the two levels were Flip and Gaussian. The crossover probability (F4) varied from 0.5 to 1.0 in steps of 0.1, the mutation probability (F5) varied from to 0.1 in steps of 0.009, and the replacement percentage (F6) varied from 0.2 to 0.5 in steps of Thus, the total number of experimental combinations was 12,096 ( ). We ran 50 simulation replications for each combination yielding a total of 604,800 runs. The response was the GA s average performance. The results of an analysis of variance (ANOVA) performed at a significance level of a = 0.05 is displayed in Table 3. The results show that all the factors were significant except for the mutation operator and a number of the interaction effects were also significant. Based on these experiments, the factors settings of

9 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) RW, replacement percentage of 0.25, UX, crossover with probability of 0.9, FM, mutation with probability of 0.01 gives the best average GA performance for problem 503. In order to investigate the sensitivity of the GA solution to different settings (factors F1 F6), we conduct a full experimental design for the moderate-size several orbit instance Fig. 5 depicts the effects of each factor on the mean fitness value. Although the percentage differences in the fitness scores in the graph are small, the absolute value difference can be significant since the actual solution values are usually in the units of 100,000. This case had the same best settings as problem 503 for the following factor settings: RW, UX, FM, and crossover probability of.9. Problem 1502 best setting for the mutation probability was.03 and for the replacement probability was 0.5. Note that from Fig. 5, there is not a significant difference in the mean fitness objective between a setting of.01 and.03 for the mutation probability. The only factor where there was a slight difference in the objective value from the best settings of problem 503 was in the replacement percentage. Overall, this analysis showed that the best settings for problem 503 hold reasonably well for another problem setting (1502). Thus, we will use these settings as the basis for all subsequent experiments. A two factor experiment is performed to investigate the effect of population size and number of generations on the GA s performance on problem For population size, we tested values of 50, 100, 150, and 200 and for the number of generations we tested values of 5000, 10,000, 20,000, 30,000, 40,000, 50,000, and 60,000 generations. Each combination was replicated 100 times; Table 4 shows the fraction of the 100 replications in which the GA was able to identify the optimal solution and Fig. 6 plots the mean fitness for each combination. The results show the superiority of a population size of 200 where 70% of the time the optimal solution was identified for a number of generations greater than 40,000. As Fig. 6 shows, the algorithm converges rapidly for a population size of Benchmark test results We tested the proposed GA on the 20 problem instances provided by the French National Space Agency (CNES) and described in detail in Bensana et al. (1999). Optimal solutions for all the single orbit problems are already known in the literature while for the multiple orbit problem only instance 1502 (209 candidate photographs and 203 constraints) is there a known optimal solution. For this set of instances, TS is the only local search technique used for modeling the selection problem for SPOT5 (Bensana et al., 1999; Vasquez and Hao, 2001). The GA heuristic was coded using the MATLAB software and tested on a Fujitsu Siemens Laptop, Intel (R) Pentium (R) M with 240 MB RAM, 40 GB HDD, 1.6 GHz speed computer system running Windows XP. The first set of experiments compares the performance of the GA against the best known results for the Max P problem and against the optimal solution as found by CPLEX for the Max ni problem since there are no published results for the Max ni problem. We note that we used the neos server for access to CPLEX and we used a stopping criterion of 36,000 CPU s for CPLEX since the neos server would not allow for a larger stopping time. The second set of experiments compares the GA in solving the multi-criteria problem of both criterions Investigating the GA s reliability For each instance, the GA was run 100 times with different random seeds to obtain its maximum, average, and minimum performance of the GA. Based on the previous experiments on Table 6 The best GA solution at various w2 values for the different SPOT5 problems. Problem code w 2 = 0.0 w 2 = 0.1 w 2 = 0.2 w 2 = 0.3 w 2 = 0.4 w 2 = 0.5 w 2 = 0.6 w 2 = 0.7 w 2 = 0.8 w 2 = 0.9 w 2 = 1.0 P ni P ni P ni P ni P ni P ni P ni P ni P ni P ni P ni , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Total 11,42, ,28, ,32, ,32, ,24, ,32, ,32, ,32, ,18, ,18, ,34,

10 518 M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010) the single orbit instance 503 and the several orbit instance 1502, the GA s parameters were set to: Roulette wheel selection. Replacement percentage of Uniform crossover with probability of 0.9. Flip mutation with probability of We note that these settings maximized the performance for only problem 503 and we use these settings for the other problem instances without conducting another full experimental design. Table 5 displays the results of our experiments on all the 20 problem instances for two objective functions: maximization of profit and number of acquired photographs. For each problem instance, the table displays the best known solution (profit) with an * indicating a known optimal solution and the best, mean, and worst solution out of the 100 GA runs for each objective value and the GA s reliability measured in terms of the fraction of times it could find the optimal/best known solution divided by 100. As the table shows, the GA was able to identify the optimal solution for all single orbit instances without capacity constraints (the first 13 instances in Table 5) and find the optimal solution for one of the several orbit instances coded It was able to match or find a higher profit for all the multiple orbit problem instances. For problem instances 1403 and 1504, the GA solution gives a higher profit of 176,139 and 126,236 than the best known solution respectively. The GA s reliability indicated by the Rel column in the table decreases with increasing problem size. However, in all problem instances, the results show that 100 replications are sufficient in identifying a suitable best solution. For the Max ni problem, the GA could find the optimal ni for the 13 single orbit instances and problems 1502 and 1504 for the multiple-orbit instances Investigating the relationship between P and ni In order to investigate the relationship between the P and ni objectives, we solve all the SPOT5 problems as a multi-criteria problem Max w 1 ni + w 2 P with w 1 = 1 w 2 and w 2 values ranging from 0.0 to 1.0 in steps of 0.1. A w 2 value of 0.0 is equivalent to solving the Max ni problem and a value of 1.0 is equivalent to solving the Max P. Table 6 gives the best solution found by the developed GA at various values of w 2 for the twenty SPOT5 problem instances. Fig. 7 plots these values for problems 408 and 412. As the plot shows, for w 2 greater than 0, the profit is roughly the same but with different ni values so the SPOT5 Company has the opportunity to have the same profit but serve as many customers as possible. For example, in problem 408, the Max P is 3082 for ni =60 and 63 photographs. These plots help to illustrate the benefit of solving the multi-criteria problem. Table 7 shows a comparison of the GA against CPLEX for maximizing the three objectives of Max P, Max ni, and Max P + ni with w 2 = 0.5. A CPLEX stopping criteria of 36,000 s is adopted and indicated by the plus (+) sign. We note that the neos server would not allow us to use a larger stopping criterion. Based on the experimentation, a stopping criterion of 36,000 is also adopted for the GA and the CPU column indicates the overall average of running each separate problem. The last column shows the difference in the solution of the multi-criteria problems between the GA and CPLEX. As the table shows, the number of acquired photographs increases when the objective changes to this measure; however, this comes at the expense of lower profit. For example, with the new objective, the number of acquired photographs increases to 35 from 31 but the profit decreases to 44 from 49 for problem instance 404. For this instance the weighted objective function has the same profit of 49 with a higher number of acquired photographs of 33 as the single objective of maximizing profit. This is repeated for problems 28, 408, 412, 503, 1401, and The results show that the weighted objective function can be an effective method for balancing these two criteria. The table shows a comparison between the GA and CPLEX with regard to CPU seconds and memory allocation with a stopping criterion of 36,000 s for both algorithms. The reason for low and fixed CPU memory allocation for the GA is the fact that it does not need memory except for allocating population size and performing the genetic operators so this amount remains constant as iteration number increases. However, the CPLEX memory increases rapidly due to increasing the search space size explored by the branch and bound technique adopted by the algorithm. The table shows that GA on the average takes a longer time to solve SPOT5 problems compared with CPLEX but it always finds solutions that are equal or better than it. 6. Conclusion and future directions In this paper, we developed a GA-based approach for solving the daily management SPOT5 scheduling problem for maximizing the daily profit, number of photographs taken, and the multi-criteria problem. The computational results on the small problem sizes show that CPLEX outperforms the GA approach, but for the larger problem sizes the GA approach is a good alternative. Although we adopt a stopping criterion of 36,000 s for both algorithms, the developed GA could match or find better solutions but it takes more computational time than CPLEX. We were able to match the best known solution for 18 of the test data sets and for two P w 2 P(L) ni(r) ni P w 2 P(L) (a) Problem 408 (b) Problem 412 Fig. 7. Double-y plot of P and ni, vs. w 2. ni(r) ni

11 Table 7 GA and CPLEX results on single and multi-criteria decision making SPOT5 benchmarks. Problem code CPLEX GA Max P + ni deviation Max P Max ni Max P + ni CPU s Memory (bytes) Max P Max ni Max P + ni CPU s Memory (bytes) Pop_size/No_gen P ni P ni P ni P ni P ni P ni a a 70 a 45 a K 70 a a 70 a 45 a K 50/ a 34 10, a a 34 a K a 34 10, a a 34 a K 50/ ,067 a , a a 80 a K a a a 80 a 15, K 200/350, a 46 55, a a 47 a M a 46 55, a a 47 a K 200/35, a a 114 a 98 a M 115 a a 114 a 98 a b 2460 K 200/10, a a 49 a 33 a K 49 a a 49 a 33 a K 200/25, a a 3082 a 63 a K 3082 a a 3082 a 63 a K 200/45, a 79 10, a a 80 a K a 79 14, a a 81 a K 200/40, a 95 11, a a 99 a M a 98 13, a a 100 a 15, K 200/35, a a 9096 a 72 a K 9096 a a 9096 a 72 a K 200/40, a a a 86 a M a a a 86 a K 200/40, a 89 10, a a 92 a K a 90 10, a a 94 a b 2472 K 200/140, a 96 12, a a 97 a M a 96 15, a a 96 a b 2732 K 200/100, a , a 148 a 22, M a , a 148 a b 3396 K 200/71, , , , b 132 M 176, , , b 5360 K 300/380, , , , b 1945 M 176, , , b 6608 K 300/260, , , , b 1229 M 176, , , b 7880 K 300/200, a , a a 166 a K a , a a 166 a K 200/50, , , a 124, b 90 M 126, , a 126, b 4600 K 300/10,00, , , , b 130 M 168, , , b 6968 K 300/300, Total 13,32, ,25, ,20, E+09 13,34, E E ,517 6E Italic indicates that the solution is better than the best known solution in the published literature. a CPLEX indicated that the solution is optimal. b A CPLEX and GA stopping criteria of 36,000 s has been adopted. M.A.A. Mansour, M.M. Dessouky / Computers & Industrial Engineering 58 (2010)